2 Answers
2

As fgrieu pointed out, the constants are defined in terms of a binary Linear Feedback Shift Register. Because LFSRs can be represented very efficiently using standard logic gates they have been used for pseudorandom number generation computers for decades. They have fallen out of favor for use directly as secure stream ciphers due to advances in cryptanalysis. But the round constants for Keccak are not being relied upon to provide all the properties of a cryptographically secure pseudorandom generator.

By defining them in this way, the designers of Keccak achieve a few things:

NIST specifically asked for functions for which the number of rounds could be extended. This definition of round constants provides an unlimited supply should more rounds need to be added later.

The constants are efficient to implement as an LFSR in hardware. You don't have to embed a table in ROM with 2304 bits of square and cube roots like SHA-2.

Because the constants work out to be mostly zeroes, an fully unrolled pipeline implementation could possibly be even a bit more efficient than a LFSR for every stage. Super-linear scaling FTW.

The designers stake out the claim that the security of Keccak is not heavily dependent on having perfectly statistically unbiased round constants. This may sound petty, but saving your peer reviewers from needless unproductive discussions in advance seems to be a direct benefit to public cryptanalysis. It seems to have worked for Keccak anyway.

This is not a rationale, and I confess that I do not quite get how we go from that to the values, but I can at least point to how the constant are derived. Quoting the Keccak Reference:

The additions and multiplications between the terms are in $\mathrm{GF}(2)$. With the exception of the value of the round constants $\mathrm{RC}[i_r]$, these rounds are identical. The round constants are given by (with the first index denoting the round number):
$$\mathrm{RC}[i_r][0][0][2^j-1] = \mathrm{rc}[j+7i_r] \text{ for all } 0\leq j \leq l,$$
and all other values of $\mathrm{RC}[i_r][x][y][z]$ are zero. The values $\mathrm{rc}[t]\in \mathrm{GF}(2)$ are defined as the output of a binary linear feedback shift register (LFSR):
$$\mathrm{rc}[t]=\left( x^t \bmod x^8+x^6+x^5+x^4+1 \right) \bmod x \text{ in } \mathrm{GF}(2)[x].$$